An exact analytical study is presented for the electrophoretic motion of two freely suspended dielectric spheres with thin electrical double layers in a quiescent, unbounded viscous fluid. The particle may differ in radius and in zeta potential at the surface, and the electric field is exerted perpendicular to their line of centers. The electrostatic and hydrodynamic governing equations are solved in the quasi-steady limit using bipolar coordinates. Corrections to Smoluchowski's equation are presented for various separation distances as well as several values of the particles' size and zeta potential ratios. The interaction between particles can be very strong when the surface-to-surface spacing approaches zero. The particle with smaller zeta potential is slowed down by the motion of the other, which is speeded up simultaneously by the motion of the first one, if the two particles have unequal zeta potentials of the same electrical sign. For two spheres of different signs in zeta potential, motions of both are enhanced by each other. As long as the two spheres have unequal zeta potentials, they rotate in company with translation in the same direction. For the case of two spheres with identical zeta potentials, there is no particle interaction. © 1989.
Keh, H. J., & Chen, S. B. (1989). Particle interactions in electrophoresis. II. Motion of two spheres normal to their line of centers. Journal of Colloid And Interface Science, 130(2), 556–567. https://doi.org/10.1016/0021-9797(89)90131-8